Cauchy-Schwarz inequality and particle entanglement

Abstract

The Glauber-Sudarshan P-representation is used in quantum optics to distinguish between semi-classical and genuinely quantum electromagnetic fields. We employ the analog of the Prepresentation to systems of identical bosons and show that the violation of the Cauchy-Schwarz inequality for the second-order correlation function is a proof of particle entanglement. The present derivation applies to any quantum system of identical bosons, with either fixed or fluctuating number of particles, provided that there is no coherence between different number states. In the light of recent experimental advances in single-particle detection, the violation of the Cauchy-Schwarz inequality may become an easily accessible entanglement probe in correlated many-body systems.Although the foundations of quantum and classical physics are much different, it is often difficult to construct a simple criterion of "quantumness" of a particular system. A good example of a non-classical behavior is, according to the Schrödinger equation, the ability of particles to exist in superpositions of quantum states. The most prominent manifestation of such superposition is the Young double-slit experiment for massive particles, which confirms their wave character. For optical waves it is the opposite -the non-classical electromagnetic field is that consisting of individual photons. The challenging question whether, and in what sense, the pulse of light is quantum, was among the key issues triggering the development of quantum optics.The problem was formalized by Glauber and Sudarshan in their studies on coherence in the context of correlation functions [1,2]. They employed the coherent states |Φ defined by the relationÊ (+) (x) |Φ = Φ(x) |Φ , whereÊ (+) (x) is the positive-frequency part of the electromagnetic fieldÊ(x), and expressed the density matrix using the so-called P-representationThe symbol DΦ denotes the integration measure over the set of complex fields Φ. The state of light is classical if the outcome of the measurement can be explained in terms of classical electromagnetic fields, which happens when the P-representation can be interpreted as a probability distribution, which means that it is normalized andfor any volume V. When the P-representation does not satisfy condition (2), the field is said to be quantum. Once the electromagnetic field is quantized, photons can be treated on a more equal foot with other particles. It is then reasonable to ask the question about correlations between individual particles and in this context the concept of particle entanglement emerges [3,4]. The possibility for particles to be entangled, which is a purely quantum phenomenon, has rather dramatic consequences. The quantumness of entanglement is underlined by the word "paradox" often used to describe some highly counter-intuitive phenomena such as the Einstein-Podolsky-Rosen (EPR) paradox [5] and the related Schrödinger's cat problem. Apart from fundamental aspects, systems of entangled particles have applications in quantum information [6], teleporta...

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